Redmond-Sun conjecture
Conjecture. (Stephen Redmond & Zhi-Wei Sun) Given positive integers and , and exponents and (with all these numbers being greater than 1), if , then between and there are always primes, with only the following ten exceptions:
- 1.
There are no primes between and .
- 2.
There are no primes between and .
- 3.
There are no primes between and .
- 4.
There are no primes between and .
- 5.
There are no primes between and .
- 6.
There are no primes between and .
- 7.
There are no primes between and .
- 8.
There are no primes between and .
- 9.
There are no primes between and .
- 10.
There are no primes between and .
See A116086 in Sloane’s OEIS for a listing of the perfect powers beginning primeless ranges before the next perfect power. As of 2007, no further counterexamples have been found past .